You could use cheats to "hire" your own nanny or babysitter by just moving someone back and forth between households, since it's so easy to do that now. Just have a neighbor come watch your kid and move them in, then after your sim gets home, move them back to their own place. It's easiest if the sim being moved lives with someone else so that the game won't constantly be asking you if you want to sell or keep your furniture every time you make a transfer.
The other way I get around this problem is just to have grandma come live with the family. If your sim doesn't already have a mom, you can go into CAS and make a baby with your sim and another random sim, then delete the random sim and age the person you created. Change the relationship so that the new sim is the parent now, and voila - grandma is coming to stay (and she has a family resemblance).
I know a lot of people don't like to use cheats, but there are so many issues with this game that sometimes you have to just to normalize things out.